A generalized beta copula with applications in modeling multivariate long-tailed data

Abstract

This work proposes a new copula class that we call the MGB2 copula. The new copula originates from extracting the dependence function of the multivariate GB2 distribution (MGB2) whose marginals follow the univariate generalized beta distribution of the second kind (GB2). The MGB2 copula can capture non-elliptical and asymmetric dependencies among marginal coordinates and provides a simple formulation for multi-dimensional applications. This new class features positive tail dependence in the upper tail and tail independence in the lower tail. Furthermore, it includes some well-known copula classes, such as the Gaussian copula, as special or limiting cases. To illustrate the usefulness of the MGB2 copula, we build a trivariate MGB2 copula model of bodily injury liability closed claims. Extended GB2 distributions are chosen to accommodate the right-skewness and the long-tailedness of the outcome variables. For the regression component, location parameters with continuous predictors are introduced using a nonlinear additive function. For comparison purposes, we also consider the Gumbel and t copulas, alternatives that capture the upper tail dependence. The paper introduces a conditional plot graphical tool for assessing the validation of the MGB2 copula. Quantitative and graphical assessment of the goodness of fit demonstrate the advantages of the MGB2 copula over the other copulas.